P-&Q-多项式结合方案的分类问题是代数组合论的主要研究目标之一，Terwilliger代数是研究此问题的重要工具。已有的工作主要从抽象代数的角度对Terwilliger代数进行刻画且仅考虑具有薄不可约T-模的P-&Q-多项式结合方案。本项目从新的视角，利用群表示论探讨q-Johnson方案不可约T-模的结构，进而掌握确定一般P-&Q-多项式结合方案薄不可约T-模的方法，并将此方法运用到扭Grassmann方案端点为1的薄不可约T-模的研究中；利用量子仿射代数的作用及“三对角对”的结构定理，刻画扭Grassamnn方案端点为1的非薄不可约T-模，期望通过对此非薄情形的研究，给出统一的方法处理薄和非薄情形的Terwilliger代数。本项目将丰富Terwilliger代数的研究方法，有助于P-&Q-多项式结合方案的分类，且非薄不可约T-模的相关成果可应用到关于编码的半定规划等研究中。

The classification of P-&Q-polynomial association schemes is one of research objects of Algebraic Combinatorics and Terwilliger algebra is the important tool for classifying P-&Q-polynomial association schemes which was mainly determined as an abstract algebra and only the cases of thin irreducible T-modules for P-&Q-polynomial association schemes were considered in present. In this project, we firstly employ group representation theory to discuss irreducible T-modules for q-Johnson scheme and further grasp the methods of determining thin irreducible T-modules by representations for general P-&Q-polynomial association schemes. Then, we apply these methods to research on the thin irreducible T-modules with endpoint 1 for the twisted Grassmann scheme. Finally, we will characterize the nonthin irreducible T-modules with endpoint 1 for the twisted Grassmann scheme by the structure theorem of tridiagonal pairs and the action of quantum affine algebra, and expect to give a unified way to treat Terwilliger algebras of nonthin and thin cases by working on this example with nonthin structure. Our project will enrich approaches of characterizing Terwilliger algebras and be helpful to the classification of P-&Q-polynomial association schemes, and also can find applications to the semi-definite programming for codes in the framework of Terwilliger algebras.